Table Of ContentHIRZEBRUCH-RIEMANN-ROCH FORMULAE ON
IRREDUCIBLE SYMPLECTIC KA¨HLER MANIFOLDS
1
0
0 MICHAEL BRITZE AND MARC A. NIEPER
2
n Abstract. Inthis articleweinvestigate Hirzebruch-Riemann-Rochformulae
a forlinebundlesonirreduciblesymplecticKa¨hlermanifolds. AsHuybrechtshas
J
shown in[11], for every irreduciblecomplex Ka¨hler manifoldX of dimension
8 2n,therearenumbersa0,a2,...,a2nsuchthatχ(L)= nk=0a2k/(2k)!qX(c1(L))k
for the Euler characteristic of a line bundle L, wherePqX : H2(X,C) → C is
] theBeauville-BogomolovquadraticformofX.
G
Using Rozansky-Witten classes similar to Hitchin and Sawon in [10], we
A obtain a formula expressing the a2k in terms of Chern numbers of X. Fur-
thermore, for the n-th generalized Kummer variety KnA (see [3]), we prove
.
h χ(L) = (n+1) q(c1(L))/2+n by purely algebro-geometric methods, where q
n
at istheformqX u(cid:0)ptoapositiv(cid:1)erationalconstant. Asimilarformulaisalready
known for the Hilbertscheme of zero-dimensional subschemes of length n on
m
aK3-surface(c.f.[7]).
[ Usingourresults,weareabletocalculateallChernnumbersofthegeneral-
izedKummervarietiesKnAforn≤5. Forn≤4theseresultswerepreviously
1
obtainedbySawon(c.f.[16]).
v
2
6
0
1
0 Contents
1
0 1. Some linear algebra 2
/ 2. Graph homology 3
h
t 2.1. The graph homology space 3
a
2.2. Operations with graphs and special graphs 5
m
2.3. Ω as an eigenvector of the operator ∂ 7
:
v 3. Rozansky-Witten classes 9
i 3.1. Rozansky-Witten classes in general 10
X
3.2. Rozansky-Witten invariants of symplectic K¨ahler manifolds 11
r
a 3.3. Examples of Rozansky-Witten classes 12
3.4. Properties of Rozansky-Witten classes 13
4. The Euler characteristic of a line bundle in terms of the quadratic form 13
4.1. The quadratic form on an irreducible symplectic manifold 13
4.2. A Hirzebruch-Riemann-Rochformula 14
5. Generalized Kummer Varieties. 17
5.1. Basic Facts 17
5.2. Explicit Hirzebruch-Riemann-Roch for KnA. 18
6. Characteristic numbers of KnA 22
References 23
We are very grateful to Daniel Huybrechts for having carefully read preliminary versions of
thispaperandtoDanielHuybrechts,ManfredLehnandmanyothersfortheirsupporttousand
helpfuldiscussionsaboutthesubject.
1
2 M. BRITZE AND M. A. NIEPER
1. Some linear algebra
Inthis section,let k denote a field ofcharacteristiczero,V a k-vectorspace and
A a k-algebra.
We identify the exterior algebra (V∗) A with the space n Altr(V,A) of
⊗k r=0
alternating multilinear forms on V with values in A as vector spaces by setting
V L
(1.1) ((α α ) a)(v ,...,v )=det((α (v )) ) a
1 r 1 r i j ij
∧···∧ ⊗ ·
for α ,...,α V∗, v ,...,v V and a A.
1 r 1 r
∈ ∈ ∈
Definition 1. We call an element σ 2(V∗) a symplectic form on V and V
∈
together with σ a symplectic k-vector space if the map
V
(1.2) I :V V∗,v σ(,v)
σ
→ 7→ ·
is an isomorphism of vector spaces.
Remark. A symplectic vector space is always of even dimension.
If σ is a fixed symplectic form on V, we will identify V and V∗ using the iso-
morphismI . Inparticular,we haveaninduced dualsymplectic formσ˜ 2V on
σ
∈
V∗.
V
Forthe restofthis section, letσ be a fixedsymplectic formona2n-dimensional
vector space V.
Definition 2. We define a non-singular pairing
(1.3) , : V∗ A V∗ A A
k k k
h· ·i ⊗ ⊗ ⊗ →
(cid:16)^ (cid:17) (cid:16)^ (cid:17)
by setting
(1.4) (α α ) a,(β β ) b =δ det((σ∗(α ,β )) ) ab
1 r 1 s rs i j ij
h ∧···∧ ⊗ ∧···∧ ⊗ i ·
for α ,...,α ,β ,...,β V∗ and a,b A, where δ is Kronecker’s δ.
1 r 1 s rs
∈ ∈
ThefollowingpropositionwillbeusedlaterinthecasethatV isatangentspace
of a complex manifold at a point p and A is the algebra of anti-holomorphic forms
at p.
Proposition 1. LetσbeasymplecticformonV and : (V∗) A 2n(V∗)
k k
⊗ → ⊗
A the canonical projection onto the forms of top degree.
R V V
For every α (V∗) we have
∈
V
(1.5) (α expσ)= ( α,expσ expσ).
∧ h i
Z Z
Proof. We can assume that A=k and α=α α 2p(V∗) with p N ,
1 2p 0
∧···∧ ∈ ∈
α V∗. So we have to prove
i
∈ V
(n p)!
α2p σ(n−p) = − α,σp σn.
∧ p! n! h i
·
Let e ,...,e be a symplectic basis of V and ϑ1,...,ϑ2n the corresponding dual
1 2n
basis of V∗, i.e. σ = n ϑ2i−1 ϑ2i and σ˜ = n e e . It follows that
i=1 ∧ i=1 2i−1 ∧ 2i
P P
HRR ON AN IRREDUCIBLE SYMPLECTIC KA¨HLER 3
σn =n! ϑ1 ϑ2n. It is
· ∧···∧
α σ(n−p),σn
∧
D E α α ϑ2i1−1 ϑ2ii ...
1 2p
=n!(n p)! ∧···∧ ∧ ∧ ∧
− (cid:10) ϑ2in−p−1 ϑ2in−p,ϑ1 ϑ2n
1≤i1<·X··<in−p≤n ···∧ ∧ ∧···∧
=n!(n p)! α α ,ϑ2j1−1 ϑ2j1 ϑ2jp−1(cid:11) ϑ2jp
1 2p
− ∧···∧ ∧ ∧···∧ ∧
1≤j1<X···<jp≤n(cid:10) (cid:11)
n!(n p)!
= − α,σp .
p! h i
Since σn spans 2n(V∗) and σn,σn =n!2, this proves the proposition.
h i
Remark. HitchinVand Sawon have stated this result for α being a two-formin [10],
p. 8. Note that they identify the exterior algebra over V∗ with the alternating
forms on V in a different way than we do.
Extending their formula to arbitrary degree of α is crucial for this work.
Definition 3. ForeverysetS,wedenoteby S theGraßmannalgebragenerated
k
by the elements of S over k. If S′ is a subset of S, we view (S′) canonically as
a subalgebra of S. We denote by SymnSVthe n-th symmektric product of the
k k V
k-vector space spanned by the elements of S.
V
2. Graph homology
2.1. The graph homology space. In this article, graph means a collection of
vertices connected by edges, i.e. every edge connects two vertices. We want to call
a half-edge (i.e. an edge together with an adjacent vertex) of a graph a flag. So,
everyedgeconsistsofexactlytwoflags. Everyflagbelongs toexactlyone vertexof
the graph. On the other hand, a vertex is given by the set of its flags. It is called
univalent, if there is only one flag belonging to it, and it is called trivalent, if there
are exactly three flags belonging to it. We shall identify edges and vertices with
the set of their flags.
Agraphiscalledvertex-oriented if,foreveryvertex,acyclicorderingofits flags
is fixed.
Definition 4. A Jacobi diagram is a vertex-oriented graph with only uni- and
trivalent vertices. A connected Jacobi diagram is a Jacobi diagram which is con-
nectedasagraph. AtrivalentJacobidiagram isaJacobidiagramwithnounivalent
vertices.
We define the degree of a Jacobi diagram to be the number of its vertices. It is
always an even number.
Example. The empty graph is a Jacobi diagram,denoted by 1. The unique Jacobi
diagram consisting of two univalent vertices (which are connected by an edge) is
denoted by ℓ.
We can always draw a Jacobi diagram in a planar drawing so that it looks like
a planar graph with vertices of valence 1, 3 or 4. Each 4-valent vertex has to be
interpreted as a crossing of two non-connected edges of the drawn graph and not
as one of its vertices. Further, we want the counter-clockwise ordering of the flags
at each trivalent vertex in the drawing to be the same as the givencyclic ordering.
n
In drawn Jacobi diagrams, we also use a notation like for a part of a
···−···
graph which looks like a long line with n univalent vertices (“legs”)attached to it,
4 M. BRITZE AND M. A. NIEPER
for example ... ...for n=3. The position of n indicates the placementof the
⊥⊥⊥
legs relative to the “long line”.
Definition 5. Let S be a totally ordered set with n elements. We set
(2.1) ε(S):=s s S.
1 n
∧···∧ ∈ Q
Here, s ,...,s are the elements of S in increasin^g order.
1 n
Definition 6. Let S be a cyclicly ordered set with an odd number n of elements.
We set
(2.2) ε(S):=s s S.
1 n
∧···∧ ∈ Q
Here, s ,...,s are the elements of S in an order^compatible with the given cyclic
1 n
one. The definition of ε(S) does not depend on this order as long as the compati-
bility condition is fulfilled.
Example. If t is a trivalentvertex in a Jacobidiagram,it makes sense to write ε(t)
because we have said that we identify a vertex with the set of the flags belonging
to it.
Definition 7. LetΓ be aJacobidiagramwithk trivalentandl univalentvertices,
so m:= 3k+l is the number of its edges. Let F be the set of its flags, E the set of
2
its edges, T the set of its trivalent vertices and U the set of its univalent vertices.
A choice of total orderings of the sets T, U and every set e E (recall that
∈
an edge is identified with the set of the two flags belonging to it) are said to be
compatible with the orientation of Γ if the equality
(2.3) ε(t ) ε(t ) ε(u ) ε(u )=ε(e ) ε(e )
1 k 1 l 1 m
∧···∧ ∧ ∧···∧ ∧···∧
holds in the Graßmann algebra F generated by the elements of F. Here,
Q
t ,...,t are the elements of T in increasing order, u ,...,u are the elements
1 k 1 l
V
of U in increasing order, and e ,...,e are all the edges in arbitrary order.
1 m
Definition 8. Wedefine tobetheQ-vectorspacespannedbyallJacobidiagrams
B
modulo the IHX relation
(2.4) =
D E−F
and the anti-symmetry (AS) relation
(2.5) + =0,
I J
which can be applied anywhere within a diagram. (For this definition see also [2]
and [17].)
Furthermore, let ′ be the subspace of spanned by all Jacobi diagrams not
B B
containing ℓ as a component, and let t be the subspace of ′ spanned by all
B B
trivalent Jacobidiagrams. All these are gradedand double-graded. The grading is
induced by the degree of the Jacobi diagrams, the double-grading by the number
of univalent and trivalent vertices.
The completion of (resp. ′, resp. t) with respect to the grading will be
B B B
denoted by ˆ(resp. ˆ′, resp. ˆt).
B B B
All these spaces are called graph homology spaces and their elements are called
graph homology classes or graphs for short.
Remarks. The subspaces of ˆ spanned by the Jacobi diagrams of degree k are
k
B B
always of finite dimension. The subspace is one-dimensional and spanned by
0
B
the graph homology class of the empty diagram 1.
HRR ON AN IRREDUCIBLE SYMPLECTIC KA¨HLER 5
n
Example. If γ is a graph which has a part looking like , it will become
n ···−···
( 1)nγ if we substitute the part by due to the anti-symmetry
− ···−··· ···−n ···
relation.
2.2. Operations with graphs and special graphs.
Definition 9. Disjoint union of Jacobi diagrams induces a bilinear map
(2.6) ˆ ˆ ˆ,(γ,γ′) γ γ′.
B×B →B 7→ ∪
By mapping 1 Q to 1 ˆ, the space ˆ becomes a graded Q-algebra, which
∈ ∈ B B
hasno componentsinodddegrees. Often, weomitthe productsign“ ”. , ′, t
∪ B B B
and so on are subalgebras.
Definition 10. Letk N. WecallthegraphhomologyclassoftheJacobidiagram
∈
2k
the 2k-wheel w , i.e. w = , w = and so on. It has 2k univalent and 2k
2k 2 4
(cid:13) + #
trivalent vertices. Further, we set w :=1.
0
k1
For k ,k N, we call the graph homology class of the Jacobi diagram a
1 2
∈ kC2
double-wheel, denoted by w . In particular, w = .
k1,k2 0,0 C
Next, we define a veryspecialelement Ω ofthe graphhomologyspace with very
remarkable properties, which have been proven by Bar-Natan, Le and Thurston
(see [17]).
Definition 11. Let the series (b2k)k CN0 of the modified Bernoulli numbers
∈
(c.f. [17]) be defined by
∞ 1 sinhx
(2.7) b x2k = ln 2.
2k 2 x
k=0 2
X
Let Ω ˆ′ be the image of the element
∈B
∞
(2.8) exp b x C[[(x ) ]]
2k 2k 2k k∈N
!∈
k=1
X
under the morphism C[[(x )]] ˆ′ C of C-algebras that maps x to w .
2k Q 2k 2k
For any µ C we set Ω(µ) :→= B ∞⊗ Ω µk, where Ω is the homogeneous com-
∈ k=0 k k
ponent of degree 2k of Ω. (Note that Ω =0 for odd k.)
k
P
The following remark is stated in [17].
Remark. The modified Bernoulli numbers are connected to the usual Bernoulli
numbers B ,B ,B ,... via
1 2 3
B
2k
(2.9) b =
2k
4k(2k)!
for all k N. In addition to this, b =0.
0
∈
The generating function of the (usual) Bernoulli numbers is given by
∞
B t
(2.10) ktk = .
k! et 1
k=0 −
X
Note that B =0 for k >1 and k odd. Furthermore, B =1 and B = 1.
k 0 1 −2
Let Γ be a Jacobi diagram and u,u′ be two different univalent vertices of Γ. At
least one of them should not belong to a component ℓ of Γ. Let v (resp. v′) be
the vertex u (resp. u′) is attached to. The process of glueing the vertices u and u′
means to remove u and u′ together with the edges connecting them to v resp. v′
6 M. BRITZE AND M. A. NIEPER
andto addanew edgebetweenv andv′. Thus,wearriveatanew graphΓ/(u,u′),
whose number of trivalent vertices is the numbers of trivalent vertices of Γ and
whose number of univalent vertices is the numbers of univalent vertices of Γ minus
two. To make it a Jacobi diagram we define the cyclic orientation of the flags at v
(resp. v′) to be the cyclic orientation of the flags at v (resp. v′) in Γ with the flag
belonging to the edge connecting v (resp. v′) with u (resp. u′) replaced by the flag
belonging to the added edge.
For example, glueing the two univalent vertices of w leads to the graph .
2
C
Ofcourse,theprocessofglueingtounivalentverticesgivenabovedoesnotwork
if u and u′ are the two univalent vertices of ℓ, thus our assumption on Γ.
Definition 12. Let Γ,Γ′ be two Jacobi diagrams, at least one of them without ℓ
as a component and U = u ,...,u resp. U′ the sets of their univalent vertices.
1 n
{ }
We define
(2.11) Γˆ(Γ′):= (Γ Γ′)/(u ,f(u ))/.../(u ,f(u )),
1 1 n n
∪
f:UX֒→U′
injective
viewed as an element in ˆ.
This induces for everyBγ ˆa ˆt-linear map
∈B B
(2.12) γˆ : ˆ′ ˆ′,γ′ γˆ(γ′).
B →B 7→
Example. Set∂ := 1ℓˆ. Itisisanendomorphismusof ˆ′ ofdegree 2. Forexample,
2 B −
∂ = . By setting
+ C
(2.13) ∂(γ,γ′):=∂(γ γ′) ∂(γ) γ′ γ ∂(γ′)
∪ − ∪ − ∪
for γ,γ′ ˆ′, we have the following formula for all γ ˆ′:
∈B ∈B
n n
(2.14) ∂(γn)= ∂(γ)γn−1+ ∂(γ,γ)γn−2.
1 2
(cid:18) (cid:19) (cid:18) (cid:19)
Acting by ∂ on a Jacobi diagram means to glue two of its univalent vertices in
all possible ways, acting by ∂(, ) on two Jacobi diagrams means to connect them
· ·
by glueinganunivalentvertexofthe firstwith anunivalent vertexofthe secondin
all possible ways. For example, we have
2k−2
(2.15) ∂(w )=k w
2k n,2k−2−n
n=0
X
for k N and, by the IHX relation,
∈
(2.16) ∂(w ,w )=8k k w
2k1 2k2 1 2 2k1−1,2k2−1
for k ,k N.
1 2
∈
Definition 13. Let Γ,Γ′ be two Jacobi diagrams, at least one of them without ℓ
as a component, and U = u ,...,u resp. U′ the sets of their univalent vertices.
1 n
{ }
We define
(2.17) Γ,Γ′ := (Γ Γ′)/(u ,f(u ))/.../(u ,f(u )),
1 1 n n
h i ∪
f:UX→U′
bijective
viewed as an element in ˆt.
This induces a ˆt-bilinBear map
B
(2.18) , , ˆ′ ˆ ˆt,
h· ·i B ×B →B
which is symmetric on ˆ′ ˆ′.
B ×B
HRR ON AN IRREDUCIBLE SYMPLECTIC KA¨HLER 7
Notethat Γ,Γ iszerounlessΓandΓ′ haveequalnumbersofunivalentvertices.
h i
In this case, the expression is the sum over all possibilities to glue the univalent
vertices of Γ with univalent vertices of Γ′.
Proposition 2. The map 1, : ˆ ˆt is the canonical projection map, i.e. it
h ·i B → B
removes all non-trivalent components from a graph. Furthermore, for γ ˆ′ and
γ′ ˆ, we have ∈ B
∈B
ℓ
(2.19) γ, γ′ = ∂γ,γ′ .
2 h i
(cid:28) (cid:29)
For γ,γ′ ˆ, we have the following (combinatorial) formula:
∈B
(2.20) exp(∂)(γγ′),1 = exp(∂)γ,exp(∂)γ′
h i h i
Proof. The formula (2.19) should be clear from the definitions.
Let us investigate (2.20) a bit more. We can assume that γ and γ′ are Jacobi
diagrams with l resp. l′ univalent vertices and l+l′ =2n with n N . So we have
0
∈
to prove
∂n ∞ ∂m ∂m′
(γγ′)= γ, γ′ ,
n! m! m′!
m,Xm′=0 * +
l−m=l′−m′
since ,1 : ˆ ˆt means to remove the components with at least one univalent
h· i B → B
vertex. Recallingthe meaningof , ,itshouldbe clearthat(2.2)followsfromthe
h· ·i
fact that applying ∂k on a Jacobi diagram means to glue k pairs of its univalent
k!
vertices in all possible ways.
2.3. Ω as an eigenvector of the operator ∂. As said above, the element Ω
playsacentralroleinthe“WheelingTheorem”(see[17], whereBar-Natan,Leand
Thurston have gave a knot theoretical prove of this theorem). Hitchin and Sawon
([10])discoveredthatthistheoremtogetherwiththeideasofRozanskyandWitten
([15]) can be used to deduce some interesting facts about characteristic classes on
irreducible symplectic K¨ahler manifolds.
ToproveourHirzebruch-Riemann-RochformulaonirreduciblesymplecticK¨ahler
manifolds alongthe ideas ofHitchin andSawon,we could makeuse of the “Wheel-
ing Theorem”. However, it tells us a lot more about Ω than we actually need. In
fact,weonlyneedthe statementgivenintheorem2.1aboutΩtoproveourresults.
Similarly, everything stated in [10] that is based on the “Wheeling Theorem” can
also be based on theorem 2.1.
The following theorem is a corollary of lemma 6.2 in [17]. Nevertheless, we give
another proof here, which does not use any knot theory.
Theorem 2.1. For each µ C, the graph Ω(µ) is an eigenvector of the endomor-
∈
phismus ∂ : ˆ′ ˆ′ with the eigenvalue µ2 , i.e.
B →B 48C
µ2
(2.21) ∂Ω(µ)= Ω(µ).
48C
Tomaketheproofofthetheoremreadable,wefirststatetwolemmata. Thefirst
lemma is a combinatorial one, which makes use of the IHX relations. It collects
some results of Chmutov, Dasbach and Duzhin Dasbach (see [5] and [6]).
Lemma 2.2. Let be the subspace of ˆ that is spanned by all graphs w with
i,j
W B
i,j N . Let P : Sym3((x ) ) be the map defined by
∈ 0 W → Q n n∈N0
2 ∞ ( 1)l+m i j x x x for i+j even
(2.22) P(w )= l,m=0 − l m l m i+j−l−m
i,j
(0P (cid:0)(cid:1)(cid:0) (cid:1) for i+j odd.
Then P is injective.
8 M. BRITZE AND M. A. NIEPER
Proof. First, we show that P is well-defined.
Foreveryn N ,let bethesubspaceof spannedbyallw withi+j =n.
0 n i,j
∈ W W
It is enough to show that for all n N there exists a injective map P :
0 n n
Sym3((x ) ) that fulfills (2.22).∈Further, we can restrict ourselves to thWe ca→se
Q n n∈N0
of even n due to lemma 6.2 of [6], which says that w is homologous to zero for
i,j
odd n. (This follows at once from the anti-symmetry relation.)
For any N > n, the map P defined in section 3.1 of [5] can be taken as P
gl(N)
when restricted to . This is because of proposition 4.5 of [5], where it is shown
n
W
that P evaluated at w equals the right hand side of (2.22).
gl(N) i,j
It remains to show that P is injective. This can be proven by a dimension
n
argument: By lemma 6.2 and lemma 6.8 of [6] the image of under P has at
n n
W
least the dimension of , so P is injective.
n n
W
Lemma 2.3. Let B ,B ,B ,... denote the Bernoulli numbers. The following for-
0 1 2
mula holds in Sym3((x ) ):
Q n n∈N0
∞ k−2 ∞
B n k 2 n
(2.23) k ( 1)l+m − − xx x
l m k−2−l−m
k! − l m
k=2 n=0l,m=0 (cid:18) (cid:19)(cid:18) (cid:19)
X X X
∞ ∞
B B i 1 j 1 1
+ i j ( 1)l+m − − x x x = x3.
i! j! − l m l m i+j−2−l−m 12 0
i,j=2 l,m=0 (cid:18) (cid:19)(cid:18) (cid:19)
X X
Proof. In Q[X ,X ,X ], we calculate
1 2 3
∞ k−2 ∞
B n k 2 n
k ( 1)l+m − − Xl Xm Xk−2−l−m
πX∈S3 kX=2 k! nX=0l,Xm=0 − (cid:18)l(cid:19)(cid:18) m (cid:19) π(1) π(2) π(3)
∞ ∞
B B i 1 j 1
+ i j ( 1)l+m − − Xl Xm Xi+j−2−l−m
i,j=2 i! j! l,m=0 − (cid:18) l (cid:19)(cid:18) m (cid:19) π(1) π(2) π(3) !
X X
∞ k−2
B
k n k−2−n
= X X X X
k! π(3)− π(1) π(3)− π(2)
πX∈S3 kX=2 nX=0(cid:0) (cid:1) (cid:0) (cid:1)
∞
B B
+ i j(X X )i−1(X X )j−1
i! j! π(3)− π(1) π(3)− π(2) !
i,j=2
X
∞
1 Bk k
X X
X X k! π(3)− π(2)
1 π(3)− π(2) k=2
= X (cid:0) (cid:1)
πX∈S3 Xπ(1)−Xπ(2) − Xπ(3) 1 Xπ(1) ∞ Bkk! Xπ(3)−Xπ(1) k
1 − Xk=2 (cid:0) (cid:1)
+
(X X )(X X )
π(3) π(1) π(3) π(2)
− −
∞ ∞
Bk k Bk k
X X X X
· k! π(3)− π(1) ! k! π(3)− π(2) !!
k=2 k=2
X (cid:0) (cid:1) X (cid:0) (cid:1)
1 1
= 1 exp Xπ(3)−Xπ(2) −1 − Xπ(3)−Xπ(2)
X X 1 1
πX∈S3 π(1)− π(2) −(cid:0)exp X (cid:1)X 1 + X X
π(3)− π(1) − π(3)− π(1)
1 (cid:0) 1 1 (cid:1)
+ +
exp Xπ(3)−Xπ(1) −1 − Xπ(3)−Xπ(1) 2!
(cid:0) (cid:1)
HRR ON AN IRREDUCIBLE SYMPLECTIC KA¨HLER 9
1 1 1 1
+ = .
· exp Xπ(3)−Xπ(2) −1 − Xπ(3)−Xπ(2) 2!! 2
(cid:0) (cid:1)
This proves the lemma because there is a well-defined Q-linear map
Sym3((x ) ) Q[X ,X ,X ],x x x Xi Xj Xk ,
Q n n∈N0 → 1 2 3 i j k 7→ π(i) π(2) π(3)
πX∈S3
which is injective.
Proof of the theorem. Since ∂ is a linear operator of degree 2 on ˆ′ and Ω(µ) =
∞ Ω µ2k, we can assume that µ=1. Set Γ:= ∞ b −w . WBe have
k=0 2k k=1 2k 2k
P ∞ ∂(Γn) P
∂Ω=∂exp(Γ)=
n!
n=0
X
∞ Γn−1 ∞ ∂(Γ,Γ) Γn−2
= ∂(Γ) +
(n 1)! 2 (n 2)!
n=1 − n=2 −
X X
1
= ∂Γ+ ∂(Γ,Γ) Ω,
2
(cid:18) (cid:19)
so we see that Ω is an eigenvector. We just have to calculate the eigenvalue:
∞ ∞
1 ∂(w ,w )
2i 2j
∂Γ+ ∂(Γ,Γ)= b ∂w + b b
2k 2k 2i 2j
2 2
k=1 i,j=1
X X
∞ 2k−2 ∞
= kb w + 4ijb b w
2k n,2k−2−n 2i 2j 2i−1,2j−1
k=1 n=0 i,j=1
X X X
∞ 2k−2 ∞
1 B 1 B B
2k 2i 2j
= w + w
n,2k−2−n 2i−1,2j−1
4 (2k)! 4 (2i)!(2j)!
k=1 n=0 i,j=1
X X X
∞ k−2 ∞
1 B 1 B B
k i j
= w + w .
n,k−2−n i−1,j−1
4 k! 4 i! j!
k=2 n=0 i,j=2
X X X
Applying the (injectiv) map P of lemma 2.2 and using lemma 2.3 yields
∞ k−2 ∞
1 1 B B B
k i j
P ∂Γ+ ∂(Γ,Γ) = P w + w
n,k−2−n i−1,j−1
2 4 k! i! j!
(cid:18) (cid:19) k=2 n=0 i,j=2
X X X
1 1 1
= x3 =P w =P .
24 0 48 0,0 48C
(cid:18) (cid:19) (cid:18) (cid:19)
Because of the injectivity of P, this proves the theorem.
3. Rozansky-Witten classes
The idea to associate to every graph Γ and every hyperk¨ahler manifold X a
cohomology class RW (Γ) is due to Rozansky and Witten (c.f. [15]). Kapranov
X
showedin[12]thatthemetricstructureofahyperk¨ahlermanifoldisnotnessessary
to define these classes. It was his idea to build the whole theory upon the Atiyah
classandthe symplecticstructureofasymplectic K¨ahlermanifold. We willusehis
definition of Rozansky-Witten classes in this section.
10 M. BRITZE AND M. A. NIEPER
3.1. Rozansky-Witten classes in general. Let k be a field of characteristic
zero, V a finite-dimensional k-vector space, A = ∞ A a Z-graded (super-
i=0 i
)commutative k-algebra and σ a symplectic form on V.
L
We shall use this general setting later in the case that V = is the holo-
X,x
T ∗
morphic tangent space of a complex manifold X at a point x and A=Ω is the
X,x
Graßmann algebra of anti-holomorphic forms at x.
For every Jacobi diagram Γ with k trivalent and l univalent vertices and every
α Sym3V A , we define an element
k 1
∈ ⊗
l
(3.1) RW (Γ) (V∗) A
σ,α k
∈ ⊗
by the following procedure: ^
Let T denote the set of trivalent vertices, U the set of univalent vertices, E the
set of edges and F the set of flags of Γ. So, l = U and k = T . The Jacobi
| | | |
diagram Γ defines the map
(3.2)
ΦΓ :(Sym3V A )⊗T (EndV)⊗U (1) V⊗t A⊗T V⊗U (V∗)⊗U
⊗ 1 ⊗ −→ !⊗ 1 ⊗ ⊗
t∈T
O
(2) V⊗F (V∗)⊗U A⊗T
−→ ⊗ ⊗ 1
(3) V⊗e (V∗)⊗U A⊗T
−→ ⊗ ⊗ 1
e∈E
O
(4) (V⊗2)⊗3k2+l (V∗)⊗l (A1)⊗k
−→ ⊗ ⊗
(5) l(V∗) A ,
k
−→ ⊗
where ^
1. is induced by the inclusions of the symmetric tensors Sym3V in the spaces
V⊗t witht T (noteagainthattisasetofthreeelements)andthecanonical
∈
identification EndV =V V∗,
⊗
2. is induced by the canonicalbijection U t F which is on eacht T
∐ t∈T → ∈
the inclusion of the subset t in F and which maps each u U to the flag
` ∈
belonging to u,
3. isinducedbythe associativityofthe tensorproduct(note that e=F),
e∈E
4. is induced by choosing total orderings of the set E, the sets e E, the set U
∈`
andthe set T,which are compatible with the orientationofthe graphΓ, and
5. isgivenby((v v ) (v v )) (α α) (a a )
1 2 3k+l−1 3k+l 1 l 1 k
⊗ ⊗···⊗ ⊗ ⊗ ⊗···⊗ ⊗ ⊗···⊗
σ(v ,v ) σ(v ,v ) (α α ) (a a ).
1 2 3k+l−1 3k+l 1 l 1 k
7→ ····· · ∧···∧ ⊗ ·····
The map ΦΓ is independent of the specific choice made in (4). One defines
(3.3) RW (Γ):=ΦΓ(α⊗T (id )⊗U).
σ,α V
⊗
Thefollowingpropositionsummarizessomeofthe propertiesofthe mapsΦΓ we
need later on. All of them follow directly from the definitions.
Proposition 3. Identifying theset oftheunivalentvertices ofℓwiththeset 1,2 ,
{ }
we have
2
(3.4) Φℓ :(EndV)⊗2 (V∗),(v λ ) (v λ ) σ(v ,v ) (λ λ )
1 1 2 2 1 2 1 2
→ ⊗ ⊗ ⊗ 7→ · ∧
for all v ,v V, λ ,λ ^V∗.
1 2 1 2
∈ ∈
Let k N . Identifying both the set of the univalent vertices and the set of
0
∈
the trivalent vertices of the wheel w (viewed as a Jacobi diagram) with the set
2k
